Interpreting category-valued presheaves

Interpreting category-valued presheaves

[Redi Haderi]

Dear categorists,

I could not find an answer to the following, so I need your help:

Given a category C we know that the category of set-valued presheaves on C
may be interpreted as a free cocompletion of C. More precisely, if we
denote PSh(C) the presheaf category, then colimit-preserving functors from
PSh(C) to a cocomplete category D correspond to functors from C to D (via
Yoneda extension).

I am interested in an interpretation of category-valued presheaves. Is
there some sort of 2-categorical version of the above fact?

Best regards,
Redi Haderi


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Re: Interpreting category-valued presheaves

[Ross Street]

Dear Redi Haderi

Max Kelly's book ``Basic concepts of enriched category theory'' generalises
this whole Kan business to V-categories for a decent base V.
So, for the case V = Cat, you obtain the result for 2-categories.
I say ``the result'', but of course, for 2-categories there are versions for
weaker structures (bicategories, pseudofunctors, and so on).
The Cat-enriched version is the strict case which you seem to want.

Regards,
Ross Street

On 16 Feb 2020, at 6:11 AM, Redi Haderi <haderiredi@gmail.com<mailto:haderiredi@gmail.com>> wrote:

Given a category C we know that the category of set-valued presheaves on C
may be interpreted as a free cocompletion of C. More precisely, if we
denote PSh(C) the presheaf category, then colimit-preserving functors from
PSh(C) to a cocomplete category D correspond to functors from C to D (via
Yoneda extension).


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Interpreting category-valued presheaves

[Ross Street]

Dear Keith

Feeling sure that the quasicategory people would have such a result, I asked Alexander Campbell who kindly replied:

``I don't remember a reference for bicategories, but references for quasi-categories are easier to locate: Theorem 5.1.5.6 (on page 345) of Lurie's book 'Higher topos theory' and Theorem 6.3.13 (on page 288) of Cisinski's book 'Higher categories and homotopical algebra' both state that the functor (y_A)^* :Cocts(Psh(A),C) --> Fun(A,C) is an equivalence for any small simplicial set A and cocomplete quasi-category C. ''

Now back to me:
As far as bicategories are concerned, the fact that the 2-category Hom(C^{op},Cat) of pseudofunctors C^{op}\to Cat
is the free cocompletion up to biequivalence of the small bicategory C is a  fairly routine generalization of the category case.
The basic ingredient is the notion of (bicategorical) colimit colim(J,F) of  a pseudofunctor F : C \to K
weighted by a pseudofunctor J : C^{op}\to Cat; see
[14. Fibrations in bicategories, Cahiers de topologie et g\'eom\'etrie diff\'erentielle 21 (1980) 111--160]
The extension of F to a weighted colimit preserving pseudofunctor is
colim(-,F) : Hom(C^{op},Cat) \to K when K has small weighted colimits.

Also of interest, a few years ago Ren\'e Guitart pointed out a lax cocompletion property of a familiar construction
(not quite presheaves or prestacks) in his talk
``Sur le foncteur diagramme'' <http://archive.numdam.org/article/CTGDC_1973__14_2_153_0.pdf>.

Ross

On 20 Feb 2020, at 7:16 AM, Keith Harbaugh <keith.harbaugh@gmail.com<mailto:keith.harbaugh@gmail.com>> wrote:

Dear Ross, can you recommend a good source for the extension of this result  to the setting of weaker notions of categories and functors?


On Wed, Feb 19, 2020, 13:35 Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:
Dear Redi Haderi

Max Kelly's book ``Basic concepts of enriched category theory'' generalises
this whole Kan business to V-categories for a decent base V.
So, for the case V = Cat, you obtain the result for 2-categories.
I say ``the result'', but of course, for 2-categories there are versions for
weaker structures (bicategories, pseudofunctors, and so on).
The Cat-enriched version is the strict case which you seem to want.


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Re: Interpreting category-valued presheaves

[Keith Harbaugh]

Is not the result you seek (4.56) of Kelly's Basic Concepts of Enriched
Category Theory, in the case where $\mathcal V = \mathbf Cat$ ?


On Sun, Feb 16, 2020, 11:33 Redi Haderi <haderiredi@gmail.com> wrote:

> Dear categorists,
>
> I could not find an answer to the following, so I need your help:
>
> Given a category C we know that the category of set-valued presheaves on C
> may be interpreted as a free cocompletion of C. More precisely, if we
> denote PSh(C) the presheaf category, then colimit-preserving functors from
> PSh(C) to a cocomplete category D correspond to functors from C to D (via
> Yoneda extension).
>
> I am interested in an interpretation of category-valued presheaves. Is
> there some sort of 2-categorical version of the above fact?
>
> Best regards,
> Redi Haderi
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]